So, the solution?
Launch Terminal.app and use the command
sudo mdutil -i off /Volumes/
You have to do that for each drive in turn, but at least you should only have to do it once.
You can check the status of the drive by running the command
sudo mdutil -s /Volumes/
It should give the status as "Indexing disabled."
Note: You will probably have to eject and (when applicable) shut off the external drive and reconnect it to have the status accurately displayed.